A306074 Bases in which 5 is a unique-period prime.

2, 3, 4, 6, 7, 9, 19, 24, 26, 39, 49, 79, 99, 124, 126, 159, 199, 249, 319, 399, 499, 624, 626, 639, 799, 999, 1249, 1279, 1599, 1999, 2499, 2559, 3124, 3126, 3199, 3999, 4999, 5119, 6249, 6399, 7999, 9999, 10239, 12499, 12799, 15624, 15626, 15999, 19999, 20479

Offset: 1

Author

Jianing Song, Jun 19 2018

Keywords

• easy
• nonn

Comments

A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.

A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.

b is a term if and only if: (a) b = 5^t + 1, t >= 1; (b) b = 2^s*5^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 7.

For every odd prime p, p is a a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 5, the nontrivial bases are 2, 3, 7.

Examples

			1/5 has period length 4 in base 2. Note that 3 and 5 are the only prime factors of 2^4 - 1 = 15, but 1/3 has period length 2, so 5 is a unique-period prime in base 2.
1/5 has period length 4 in base 3. Note that 2 and 5 are the only prime factors of 3^4 - 1 = 80, but 1/2 has period length 1, so 5 is a unique-period prime in base 3.
1/5 has period length 4 in base 7. Note that 2, 3 and 5 are the only prime factors of 7^4 - 1 = 2400, but 1/2 and 1/3 both have period length 1, so 5 is a unique-period prime in base 7.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..544
• Wikipedia, Unique prime

Crossrefs

Cf. A040017 (unique-period primes in base 10), A144755 (base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), this sequence (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).

Programs

PARI

p = 5;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));